Pushforward (homology) explained

f : X → Y between two topological spaces is a homomorphism f _{*} :H _{n\left(X\right)} → H _{n\left(Y\right)} between the homology groups for n \geq 0 .

Homology is a functor which converts a topological space

X into a sequence of homology groups H _{n} \left(X\right) . (Often, the collection of all such groups is referred to using the notation H _{*} \left(X\right) ; this collection has the structure of a graded ring.) In any category, a functor must induce a corresponding morphism. The pushforward is the morphism corresponding to the homology functor.

Definition for singular and simplicial homology

We build the pushforward homomorphism as follows (for singular or simplicial homology):

C _{n\left(X\right)} and C _{n\left(Y\right)} defined by composing each singular n-simplex \sigma _{X} : \Delta ^{n →} X with f to obtain a singular n-simplex of Y , f _{\#} \left(\sigma _{X\right)} = f\sigma _{X} : \Delta ^{n →} Y . Then we extend f _{\#} linearly via f _{\#} \left(\sum _{tn} _{t\sigma} _{t\right)} = \sum _{tn} _{tf} _{\#} \left(\sigma _{t\right)} .

The maps

f _{\#} : C _{n\left(X\right) →} C _{n\left(Y\right)} satisfy f _{\#} \partial = \partial f _{\#} where \partial is the boundary operator between chain groups, so \partial f _{\#} defines a chain map.

We have that

f _{\#} takes cycles to cycles, since \partial \alpha = 0 implies \partial f _{\#} \left( \alpha \right) = f _{\#} \left(\partial \alpha \right) = 0 . Also f _{\#} takes boundaries to boundaries since f _{\#} \left(\partial \beta \right) = \partial f _{\#} \left(\beta \right) .

Hence

f _{\#} induces a homomorphism between the homology groups f _{*} : H _{n\left(X\right)} → H _{n\left(Y\right)} for n\geq 0 .

Properties and homotopy invariance

Two basic properties of the push-forward are:

\left( f\circ g\right) _{*} = f _{*} \circ g _{*} for the composition of maps X\overset{g}{ → }Y\overset{f}{ → }Z .\left( id _{X} \right) _{*} = id where id _{X} : X → X refers to identity function of X and id \colon H _{n\left(X\right)} → H _{n\left(X\right)} refers to the identity isomorphism of homology groups.

A main result about the push-forward is the homotopy invariance: if two maps

f,g\colon X → Y are homotopic, then they induce the same homomorphism f _{*} = g _{*} \colon H _{n\left(X\right)} → H _{n\left(Y\right)} .

This immediately implies that the homology groups of homotopy equivalent spaces are isomorphic:

The maps

f _{*} \colon H _{n\left(X\right)} → H _{n\left(Y\right)} induced by a homotopy equivalence f\colon X → Y are isomorphisms for all n .

References

• Allen Hatcher, Algebraic topology. Cambridge University Press, and